Let $f(x) \in W^{s,2}(\Omega) \equiv H^s$, where $\Omega \subseteq \mathbb{R}^d$ and $W^{s,2}$ is a $(s,2)$-Sobolev space. Clearly, $W^{s,2}$ is an Reproducing Kernel Hilbert Space (RKHS) and therefore $f$ is a linear bounded operator (i.e. $|f(x)| \le M\left\|f\right\|$ for $M > 0$ holds).

The question is: are derivatives of $f$ also linear bounded operators (they're in $L^2(\Omega)$, but does $|\nabla_x f(x)| \le M'\left\|\nabla_x f\right\|$ hold for some $M' > 0$?)?

Thank you